Matrix representations and independencies in directed acyclic graphs
Journal article, 2009

For a directed acyclic graph, there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph. Both criteria are based on special types of path in graphs. They are called separation criteria because independence holds whenever the conditioning set is a separating set in a graph theoretical sense. We introduce and discuss an alternative approach using binary matrix representations of graphs in which zeros indicate independence statements. A matrix condition is shown to give a new path criterion for separation and to be equivalent to each of the previous two path criteria. © Institute of Mathematical Statistics, 2009.

Conditional independence

Separation criteria

Partial closure

Edge matrix

Partial inversion

Parent graph

Stepwise data generating process


Giovanni M. Marchetti

University of Florence

Nanny Wermuth

University of Gothenburg

Chalmers, Mathematical Sciences

Annals of Statistics

0090-5364 (ISSN)

Vol. 37 2 961-978

Subject Categories

Algebra and Logic

Probability Theory and Statistics

Discrete Mathematics



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